THE QFT NOTES 5 Badis Ydri Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria. December 9, 2011 Contents 1 Renormalization of QED 2 1.1 Example III: e− + µ− e− + µ− ......................... 2 −→ 1.2 Example IV : Scattering From External Electromagnetic Fields . ........ 3 1.3 One-loop Calculation I: Vertex Correction . .... 6 1.3.1 FeynmanParametersandWickRotation . 6 1.3.2 Pauli-Villars Regularization . 11 1.3.3 Renormalization (Minimal Subtraction) and Anomalous Magnetic Moment 13 1.4 Exact Fermion 2 PointFunction .......................... 16 − 1.5 One-loop Calculation II: Electron Self-Energy . ..... 18 1.5.1 ElectronMassatOne-Loop . .. .. .. 18 1.5.2 The Wave-Function Renormalization Z2 .................. 21 1.5.3 The Renormalization Constant Z1 ...................... 22 1.6 Ward-TakahashiIdentities . ... 24 1.7 One-Loop Calculation III: Vacuum Polarization . .... 28 1.7.1 The Renormalization Constant Z3 and Renormalization of the Electric Charge..................................... 28 1.7.2 Dimensional Regularization . 30 1.8 RenormalizationofQED............................... 34 1.9 ProblemsandExercises............................... 36 1 1 Renormalization of QED 1.1 Example III: e− + µ− e− + µ− −→ The most important one-loop correction to the probability amplitude of the process e− + e+ µ− +µ+ is given by the Feynamn diagram RAD2. This is known as the vertex correction −→ as it gives quantum correction to the QED interaction vertex ieγµ. It has profound observable − measurable physical consequences. For example it will lead among other things to the infamous anomalous magnetic moment of the electron. This is a generic effect. Indeed vertex correction should appear in all electromagnetic processes. Let us consider here as an example the different process ′ ′ e−(p)+ µ−(k) e−(p )+ µ−(k ). (1) −→ This is related to the process e− +e+ µ− +µ+ by the so-called crossing symmetry or substi- −→ tution law. Remark that the incoming positron became the outgoing electron and the outgoing antimuon became the incoming muon. The substitution law is essentially the statement that the probability amplitudes of these two processes can be obtained from the same Green’s function. Instead of following this route we will simply use Feynman rules to write down the probability amplitude of the above process of electron scattering from a heavy particle which is here the muon. For vertex correction we will need to add the probability amplitudes of the three Feynamn ′ ′ diagrams VERTEX. The tree level contribution (first graph) is (with q = p p and l = l q) − − 2 ′ ′ ie ′ ′ ′ ′ (2π)4δ4(k + p k p ) (¯us (p )γµus(p))(¯ur (k )γ ur(k)). (2) − − q2 µ The electron vertex correction (the second graph) is ′ 4 4 ′ 4 4 ′ ′ e d l 1 s ′ λ i(γ.l + me) µ i(γ.l + me) s (2π) δ (k + p k p )−2 4 2 u¯ (p )γ ′2 2 γ 2 2 γλu (p) − − q Z (2π) (l p) + iǫ l me + iǫ l me + iǫ ′ ′ − − − (¯ur (k )γ ur(k)). (3) × µ The muon vertex correction (the third graph) is similar to the electron vertex correction but since it will be neglected in the limit m we will not write down here. µ −→ ∞ Adding the three diagrams together we obtain 2 ′ ′ ie ′ ′ ′ ′ ′ (2π)4δ4(k + p k p ) (¯us (p )Γµ(p ,p)us(p))(¯ur (k )γ ur(k)). (4) − − q2 µ ′ ′ This is the same as the tree level term with an effective vertex ieΓµ(p ,p) where Γµ(p ,p) is − given by 4 ′ µ ′ µ 2 d l 1 λ i(γ.l + me) µ i(γ.l + me) Γ (p ,p)= γ + ie γ ′ γ γ . (5) (2π)4 (l p)2 + iǫ l 2 m2 + iǫ l2 m2 + iǫ λ Z − − e − e 2 If we did not take the limit m the muon vertex would have also been corrected in the µ −→ ∞ same fashion. The corrections to external legs are given by the four diagrams WAVEFUNCTION. We only write explicitly the first of these diagrams. This is given by 4 4 ′ ′ e d l 1 ′ ′ γ.p + m γ.l + m ′ ′ (2π)4δ4(k + p k p ) (¯us (p )γµ e γλ e γ us(p))(¯ur (k )γ ur(k)). − − q2 (2π)4 (l p)2 + iǫ p2 m2 l2 m2 λ µ Z − − e − e (6) The last diagram contributing to the one-loop radiative corrections is the vacuum polarization diagram shown on figure PHOTONVACUUM. It is given by 2 ′ ′ ie ′ ′ ′ ′ (2π)4δ4(k + p k p ) (¯us (p )γ us(p))Πµν (q)(¯ur (k )γ ur(k)). (7) − − (q2)2 µ 2 ν d4k i(γ.k + m ) i(γ.(k + q)+ m ) iΠµν (q)=( 1) tr( ieγµ) e ( ieγν ) e . (8) 2 − (2π)4 − k2 m2 + iǫ − (k + q)2 m2 + iǫ Z − e − e 1.2 Example IV : Scattering From External Electromagnetic Fields We will now consider the problem of scattering of electrons from a fixed external electro- backgr magnetic field Aµ , viz ′ e−(p) e−(p ). (9) −→ ′ The transfer momentum which is here q = p p is taken by the background electromagnetic field backgr − Aµ . Besides this background field there will also be a fluctuating quantum electromagnetic field Aµ as usual. This means in particular that the interaction Lagrangian is of the form ¯ = eψˆ γ ψˆ (Aˆµ + Aµ,backgr). (10) Lin − in µ in The initial and final states in this case are given by ~p, s in >= 2E ˆb (~p, s)+ 0 in > . (11) | ~p in | q ′ ′ ′ ′ + ~p ,s out >= 2E ′ ˆb (~p ,s ) 0 out > . (12) | ~p out | q The probability amplitude after reducing the initial and final electron states using the appro- priate reduction formulas is given by ′ ′ ′ ′ ′ ′ s ′ s < ~p s out ~ps in > = u¯ (p )(γ.p me) ′ Gα α( p ,p) (γ.p me)u (p) . (13) | − − α − − α ′ ′ ¯ Here G ′ (p ,p) is the Fourier transform of the 2 point Green’s function < 0 out T (ψˆ ′ (x )ψˆ (x)) 0 in >, α α − | α α | viz 3 4 ′ 4 ′ d p d p ′ ′ ′ ˆ ′ ¯ˆ ′ ipx+ip x < 0 out T ψα (x )ψα(x) 0 in > = 4 4 Gα ,α(p ,p) e . (14) | | Z (2π) Z (2π) By using the Gell-Mann Low formula we get ′ ′ ˆ ′ ¯ˆ ˆ ′ ¯ˆ < 0 out T ψα (x )ψα(x) 0 in > = < 0 in T ψα ,in(x )ψα,in(x)S 0 in > . (15) | | | | Now we use Wick’s theorem. The first term in S leads 0. The second term in S leads to the contribution ′ ′ 4 ˆ ′ ¯ˆ 4 ˆ ′ ¯ˆ ¯ˆ i d z < 0 in T ψα ,in(x )ψα,in(x) in(z) 0 in > = ( ie) d z < 0 in T ψα ,in(x )ψα,in(x).ψin(z)γµ Z | L | − Z | ψˆ (z) 0 in > Aµ,backgr(z) × in | ′ α α 4 ′ µ,backgr = ( ie) d z SF (x z)γµSF (z x) A (z) − − − Z ′ ′ 4 4 α α d p d p ′ µ,backgr = ( ie) 4 4 S(p )γµS(p) A (q) − Z (2π) Z (2π) ′ ′ eipx−ip x . (16) × We read from this equation the Fourier transform ′ ′ ′ α α ′ µ,backgr Gα α( p ,p) = ( ie) S(p )γµS(p) A (q). (17) − − The tree level probability amplitude is therefore given by ′ ′ ′ s ′ s µ,backgr < ~p s out ~ps in > = ie u¯ (p )γµu (p) A (q). (18) | − The Fourier transform Aµ,backgr(q) is defined by 4 µ,backgr d q µ,backgr −iqx A (x)= 4 A (q) e . (19) Z (2π) This tree level process corresponds to the Feynman diagram EXT-TREE. The background field is usually assumed to be small. So we will only keep linear terms in Aµ,backgr(x). The third term in S does not lead to any correction which is linear in Aµ,backgr(x). The fourth term in S leads to a linear term in Aµ,backgr(x) given by 3 ( ie) ′ 4 4 4 ˆ ′ ¯ˆ ¯ˆ ˆ ¯ˆ ˆ ¯ˆ − (3) d z1 d z2 d z3 < 0 in T ψα ,in(x )ψα,in(x).ψin(z1)γµψin(z1).ψin(z2)γνψin(z2).ψin(z3) 3! Z Z Z | µ ν λ,backgr γλψˆin(z3) 0 in >< 0 out T (Aˆ (z1)Aˆ (z2)) 0 in > A (z3). (20) × | | | 4 We use Wick’s theorem. For the gauge fields the result is trivial. It is simply given by the photon propagator. For the fermion fields the result is quite complicated. As before there are in total 24 contractions. By dropping those disconnected contractions which contain SF (0) we will only have 11 contractions left. By further inspection we see that only 8 are really disconnected. By using then the symmetry between the internal points z1 and z2 we obtain the four terms ′ ¯ ¯ ¯ ¯ < 0 in T ψˆ ′ (x )ψˆ (x).ψˆ (z )γ ψˆ (z ).ψˆ (z )γ ψˆ (z ).ψˆ (z )γ ψˆ (z ) 0 in > | α ,in α,in in 1 µ in 1 in 2 ν in 2 in 3 λ in 3 | ′ ′ α α = 2 S (x z )γ S (z x) trγ S (z z )γ S (z z ) − F − 1 µ F 1 − ν F 2 − 3 λ F 3 − 2 ′ ′ α α + 2 S (x z )γ S (z z )γ S (z z )γ S (z x) F − 1 µ F 1 − 2 ν F 2 − 3 λ F 3 − ′ ′ α α + 2 S (x z )γ S (z z )γ S (z z )γ S (z x) F − 3 λ F 3 − 2 ν F 2 − 1 µ F 1 − ′ ′ α α + 2 SF (x z1)γµSF (z1 z3)γλSF (z3 z2)γνSF (z2 x) .